If you know how to do it in polar I don't see how you could be confused here. You compute the determinant of the Jacobian of the coordinate transform, just like for polar coordinates (or any other coordinate transform).

In your case, the transform is exactly the one you have listed. Just compute its Jacobian and take the determinant. Done.

The interesting question is why does area or volume transformation lead to Jacobian? Or really, more generally, transformation of differential products.

The Jacobian is nothing more than the derivative. Specifically, for a $C^1$ function $f: \mathbb{R}^n \to \mathbb{R}^n$, and $x \in \mathbb{R}^n$, there is a linear operator, $A$, which is the best linear approximation of $f$ in a neighborhood of $x$. In canonical coordinates, $A$ is a matrix and the Jacobian is an explicit formula for computing this matrix. It is commonly expressed as $J(x)$ or $Df(x)$.

A fundamental fact from linear algebra is that if $A$ is diagonalizable, then it has $n$-many eigenvectors, $\{v_i\}$, which of course form a basis for $\mathbb{R}^n$ since eigenvectors are linearly independent. Each $v_i$ has an associated eigenvalue, $\lambda_i$, which represents a scaling (and possibly reflection) in the direction of $v_i$.

It follows that upon mapping a volume element of $\mathbb{R}^n$ through $A$, the entire volume is scaled by the $\Pi \lambda_i = $ det$(A)$. Of course, the assumption that $A$ is diagonalizable is merely to aid intuition. The result that the determinant of a linear operator is a quantitative measure of volume dilation holds for all finite dimensional linear operators.

As for higher dimensions, I suppose you could define dxdudvdw=Jdrdsdtdu, but it would only be meaningful if
$\displaystyle \int _{V}dxdydzdw=\int _{V}Jdrdsdtdw$
But how do you show that? How do you even define a meaningful volume in n dimensions with meaningful limits of integration, other than a "cube"?

Ok I’m quiet new on this stuff so I want to keep it simple.
I have substituted u=r and v=Θ
At the given equation in the first original picture. So Dudu turned into drdθ
Partial derivative of (x(r,θ)...
Is with respect to dr or dθ? Or both consecutively.
And what is next then?

All I need to do is to turn dA= rdrdθ but still couldn’t see the patterns